Course Notes for EE1266 Applications of fields and waves
Chapter 1: Lecture 4
1
NOTES for Transmission Lines IV
This lecture covers Chapter 6.4
1. Smith chart
2. Parametric equations
3. Use of Smith chart: input impedance, SWR
Smith Chart is a convenience way to graphically calculate a
number of important transmission line problems based on the
reflection coefficient. Although the wide availability of computer,
the Smith Chart lost its advantage in term of computation, but it
remains as a popular graphic tool to handle transmission line. So
today, we still like to go through it in detail.
1. Smith Charts, parametric equations.
The base of Smith chart is the plot of reflection coefficient Γ and
its phase. If we plot Γ in a polar coordinate, it is an area bounded
by the circle
1
≤
Γ
Note that the reflection coefficient is always less or equal to 1. We
also know that the reflection coefficient is directly related to the
impedance of the load and the characteristic impedance of the T
line (Normalized impedance).
(
)
(
)
i
r
i
r
L
L
L
j
j
jx
r
z
Z
Z
z
Γ
−
Γ
+
Γ
+
Γ
+
=
Γ
−
Γ
+
=
+
=
=
1
1
1
1
0
If we separate the above equation for the real and imaginary part,
we have
From these parametric equations, we know two things:
(
)
2
2
2
2
2
2
1
1
1
1
1
1
⎟
⎠
⎞
⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛
−
Γ
+
−
Γ
⎟
⎠
⎞
⎜
⎝
⎛
+
=
Γ
+
⎟
⎠
⎞
⎜
⎝
⎛
+
+
Γ
x
x
r
r
r
i
r
i
r
Course Notes for EE1266 Applications of fields and waves
Chapter 1: Lecture 4
2
• Given a normalized impedance, we can define two
parametric equations. The intersect of these two circle
determine the reflection coefficient.
• Given a reflection coefficient, we can determine the
normalize impedance by find the intersect of these two
circle.
What a Smith chart can do?
• Given a load and intrinsic impedance, we can determine
the reflection coefficient.
• We can use Smith chart to determine input impedance.
Here is why and how.
1. We first determine Γ, then we draw